Bochner's technique in the theory of Riemannian almost product structures

Abstract

Let (M, g) be a compact orientable Riemannian manifold with positive definite metric g and globally defined Riemannian almost product structure (P, g), where P is a tensor field of type (i.i) such that p2 = id and g(P, P) = g. With the Riemannian almost product structure (P, g) there is associated a pair of orthogonal complementary distributions &H and.&V i corresponding to the eigenvalues of P equal, respectively, to -i and I. We denote by Pj, Rijke, and gke the components of the field P, the curvature tensor, and metric g with respect to a local coordinate system x ~, x 2, .... x n. In the expression for the scalar curvature R = gikgjeRijke we replace the contravariant components of the metric tensor g by the components pij = gjep~ of the structure tensor. We call the quantity H = pikpjeRijke obtained the scalar curvature of the Riemannian almost product structure. We call R (M) = ~ Rdv and N (7~f) = IM Ndv the global scalar curvatures of the Riemannian manifold (M~ g) and the Riemannian almost product structure (P~ g). A Riemannian almost product structure (P, g) is said to be integrable if the distributions A H and A V are integrable and thus there is given on (M, g) a pair of orthogonal complementary foliations. A foliation is said to be minimal (completely umbilical or completely geodesic) if each of its leaves is a minimal (respectively, completely umbilical or completely geodesic) submanifold of (M, g). One can prove the following theorem. THEOREM i. Let (M, g) be a compact, orientable, Riemannian manifold with globally defined Riemannian almost product structure. If ~(M) < R(M), then the Riemannian almost product structure does not define two orthogonal complementary minimal foliations on (M, g). If H(M) = R(M) such foliations can only be completely geodesic. A distribution A is said to be completely geodesic [3] if for any geodesic 7 of the Riemannian manifold (M, g) tangent at one of its points x to the plane A x, 7 is an integral curve of the distribution A. In other terminology such a distribution is called flat [4] since it is characterized by the vanishing of its second fundamental form.